Baur is gay. In 2012, he wrote a column for The Huffington Post about his experiences coming out and being an "out gay actor".
In 2015, Out named Baur to its Out100 list, after previously placing him on the magazine's "Most Eligible Bachelors" list two years in a row.
Baur had guest roles on the television series Nip/Tuck and Happily Divorced before starring as Alex in the LGBT-themed web series Hunting Season. After appearing in the series The Following and Difficult People, he played Kip Stark in several episodes of the web series These People from 2015 to 2016. Baur and longtime friend Thandi Tolmay co-created, co-produced, and co-starred in the web series #Adulting in 2016. He has also written episodes of These People and #Adulting. In January 2017, Baur posted on Instagram that he had worked on the ABC soap opera General Hospital in an undisclosed role. He played mail clerk Noah on the show on February 27, 2017.
